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Properties

Label 2-145-29.28-c3-0-13
Degree $2$
Conductor $145$
Sign $0.776 - 0.630i$
Analytic cond. $8.55527$
Root an. cond. $2.92494$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.702i·2-s + 0.260i·3-s + 7.50·4-s − 5·5-s − 0.183·6-s + 0.749·7-s + 10.8i·8-s + 26.9·9-s − 3.51i·10-s + 20.4i·11-s + 1.95i·12-s + 28.8·13-s + 0.526i·14-s − 1.30i·15-s + 52.3·16-s − 85.6i·17-s + ⋯
L(s)  = 1  + 0.248i·2-s + 0.0501i·3-s + 0.938·4-s − 0.447·5-s − 0.0124·6-s + 0.0404·7-s + 0.481i·8-s + 0.997·9-s − 0.111i·10-s + 0.559i·11-s + 0.0470i·12-s + 0.616·13-s + 0.0100i·14-s − 0.0224i·15-s + 0.818·16-s − 1.22i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 - 0.630i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.776 - 0.630i$
Analytic conductor: \(8.55527\)
Root analytic conductor: \(2.92494\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :3/2),\ 0.776 - 0.630i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.96459 + 0.696703i\)
\(L(\frac12)\) \(\approx\) \(1.96459 + 0.696703i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 5T \)
29 \( 1 + (-121. + 98.3i)T \)
good2 \( 1 - 0.702iT - 8T^{2} \)
3 \( 1 - 0.260iT - 27T^{2} \)
7 \( 1 - 0.749T + 343T^{2} \)
11 \( 1 - 20.4iT - 1.33e3T^{2} \)
13 \( 1 - 28.8T + 2.19e3T^{2} \)
17 \( 1 + 85.6iT - 4.91e3T^{2} \)
19 \( 1 - 157. iT - 6.85e3T^{2} \)
23 \( 1 - 67.9T + 1.21e4T^{2} \)
31 \( 1 - 184. iT - 2.97e4T^{2} \)
37 \( 1 - 286. iT - 5.06e4T^{2} \)
41 \( 1 + 108. iT - 6.89e4T^{2} \)
43 \( 1 + 352. iT - 7.95e4T^{2} \)
47 \( 1 + 508. iT - 1.03e5T^{2} \)
53 \( 1 + 667.T + 1.48e5T^{2} \)
59 \( 1 + 367.T + 2.05e5T^{2} \)
61 \( 1 + 513. iT - 2.26e5T^{2} \)
67 \( 1 - 94.0T + 3.00e5T^{2} \)
71 \( 1 + 38.3T + 3.57e5T^{2} \)
73 \( 1 + 721. iT - 3.89e5T^{2} \)
79 \( 1 + 365. iT - 4.93e5T^{2} \)
83 \( 1 + 1.03e3T + 5.71e5T^{2} \)
89 \( 1 + 796. iT - 7.04e5T^{2} \)
97 \( 1 - 1.02e3iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.45054725576516745453364156653, −11.85146954055235277633961387748, −10.67513594276568221495133456618, −9.833955165949601445375476566734, −8.261813350099803113131794475750, −7.31923336758179348656853738149, −6.44578017515368798900884867782, −4.92524474904053088526760762831, −3.39540183106976530669982523088, −1.59464009757649104997359901346, 1.24047315129066032862278218114, 2.97863893211145720660837776092, 4.37514838713064248504465627965, 6.16278801827752945226931279604, 7.09249553575651914247692808832, 8.179903621324380910660509692492, 9.515644568050411326782948640782, 10.92703131803976247950906875980, 11.16029875725004025015457939037, 12.58747352050427769189063910003

Graph of the $Z$-function along the critical line